Homological Modular Synthesis: A Topological Framework for Autonomous Modularization of Transformer Weight Manifolds

Modern Transformer architectures have achieved remarkable performance across a wide range of tasks, yet they remain fundamentally monolithic, relying on densely interconnected parameter spaces that provide little explicit modular organization. Existing approaches to pruning, sparsification, and modularization primarily depend on optimization heuristics or activation-based analyses, offering limited theoretical justification for why coherent functional modules should emerge within learned weight spaces. In this work, we introduce Homological Modular Synthesis (HMS), a mathematically rigorous framework that establishes topology as an organizing principle for autonomous neural modularization. HMS models converged Transformer parameters as a structured Weight Manifold, endowed with geometric and topological properties that enable principled decomposition through algebraic topology. Building upon persistent homology, information geometry, and differential topology, we establish conditions under which the learned parameter space admits locally smooth manifold structure, characterize the localization of optimization dynamics on its principal strata, and develop topology-preserving modular decomposition through Topological Surgery. We further prove that persistent topological structures remain stable under admissible modular synthesis operations and introduce a distributed reconstruction framework based on balanced cover decomposition and the Mayer–Vietoris principle, providing theoretical guarantees for scalable analysis of large neural models. Collectively, these results transform persistent topology from a descriptive tool into a constructive mathematical foundation for autonomous modular synthesis, providing a unified framework for topology-guided model compression, interpretable modular Transformer architectures, and structurally informed neural optimization. Rather than viewing modularity as an artifact of optimization, HMS demonstrates that it can be understood as an intrinsic consequence of the geometric and topological organization of learned weight manifolds.